Question: Select all polynomials that are divisible by $(x+3)$. Choose all answers that apply: Choose all answers that apply: (Choice A) A $A(x)=x^3+4x^2-9$ (Choice B) B $B(x)=x^3-x^2-18$ (Choice C) C $C(x)=x^3-6x-9$ (Choice D) D $D(x)=x^3+3x+3$
The following statements are equivalent: $(x+3)$ is a factor of $p(x)$ $p(x)$ is divisible by $(x+3)$ The remainder of $\dfrac{p(x)}{x+3}$ is $0$ We can use the polynomial remainder theorem to solve this problem: For a polynomial $p(x)$ and a number $a$, the remainder on division by $x-a$ is $p(a)$. According to the theorem, the remainder when $p(x)$ is divided by $(x+3)$, which can be rewritten as $(x-({-3}))$, is equal to $p({-3})$. So to check each polynomial is divisible by $(x+3)$, we need to check if that polynomial's value at ${x=-3}$ is zero. $\begin{aligned} A({-3})&=0 \\\\ B({-3})&=-54 \\\\ C({-3})&=-18 \\\\ D({-3})&=-33 \end{aligned}$ In conclusion, the following polynomial is divisible by $(x+3)$ : $A(x)=x^3+4x^2-9$